Kirchhoff Equations with Choquard Exponential Type Nonlinearity Involving the Fractional Laplacian

In this article, we deal with the existence of non-negative solutions of the class of following non local problem {−M(∫Rn∫Rn|u(x)−u(y)|ns|x−y|2ndxdy)(−Δ)n/ssu=(∫ΩG(y,u)|x−y|μdy)g(x,u)inΩ,u=0inRn∖Ω, where (−Δ)n/ss is the n/ s-fractional Laplace operator, n≥ 1 , s∈ (0 , 1) such that n/ s≥ 2 , Ω ⊂ Rn is a bounded domain with Lipschitz boundary, M: R+→ R+ and g: Ω × R→ R are continuous functions, where g behaves like exp(|u|nn−s) as | u| → ∞. The key feature of this article is the presence of Kirchhoff model along with convolution type nonlinearity having exponential growth which appears in several physical and biological models.